Boolean Algebra

algebra boolean-algebra

Definition

Boolean Algebra

A distributive lattice $(L,\wedge ,\vee )$ is called Boolean Algebra if it has the following properties:

1. $(L,\wedge )$ is a monoid.
2. $(L,\vee )$ is a monoid.
3. Neutral element $1_{\wedge }$ for $(L,\wedge )$:

$$\exists 1_{\wedge }\in L:\forall x\in L:x\wedge 1_{\wedge }=1_{\wedge }\wedge x=x$$

4. Neutral element $0_{\vee }$ for $(L,\vee )$:

$$\exists 0_{\vee }\in L:\forall x\in L:x\vee 0_{\vee }=0_{\vee }\vee x=x$$

5. All elements $x\in L$ have a complement $x^{\prime }\in L$:

$$\forall x\in L:\exists x^{\prime }\in L:x\vee x^{\prime }=1_{\wedge }\text{and}x\wedge x^{\prime }=0_{\vee }$$